Protons are accelerated in pulses and collide with "stationary"
H2O molecules. The interaction takes place inside a region of elevated electric
potential, so charge particles are accelerated to
a detector. The energy of the protons can be controlled, as can the composition
of the target. Specifically, we will be examining runs with 4 and 1 MeV protons
colliding with H2O, D2O, and HDO. A basic schematic of the design can be found
below.

(Note: Not to scale.)

When the proton collides with the neutral target, a number of things can
happen. The neutral target could remain intact, it could disassociate, or it
could ionize. Any positively charged particles are accelerated out of the
interaction region to the detector. (After the interaction, the protons still
have a great enough kinetic energy that they travel out of the interaction
region without being detected.) At the detector, the time of flight is measured
using the timing of the proton pulses. It can be shown that the time of flight
is proportional to sqrt(mass/charge) of the positively charge fragments. This is
intuitive since one would expect massive particles to be accelerated more slowly
and thus have a greater time of flight than lighter particles. The inverse can
be said of highly charge particles compared to low charge particles.

Corrections

This method of particle detection is useful, but it has
some limitations. For instance, the detector cannot distinguish between two
particles of identical mass and charge, or two hits involving two particles of
the same mass/charge ratio. Thus, in our system, we cannot initially distinguish between
H2(+) and D(+), or the Hydrogen ions in the H2O →
H(+) + H(+) + O(+) reaction. Furthermore, the detector efficiency is only
about 35%, so sometimes events occur without being detected, or only a fraction
of the particles involved are detected. Also, more than one reaction can occur
in the same pulse, so random coincidences appear. In the random coincidences
case, for instance, we might detected an O(+) and an OH(+). Clearly, these
fragments are not the result of an ionization and disassociation of O2H(+), they
are the results of two distinct interactions. In that case, most likely we
detected the O(+) from a H2O → H(+) + H(+) +
O(+) interaction and the OH(+) from a H2O →
H(+) + OH(+). We tackle these problems with perturbation theory.

In my explanation of our perturbation theory method, I will consider a
different, less complicated target: HD. While it gets much more complicated with
isotopic water, the same general idea applies. The possible reactions involving
HD are:

HD → HD(+): Detect HD(+)

HD → H + D(+): Detect D(+)

HD → H(+) + D: Detect H(+)

HD → H(+) + D(+): Detect H(+) and D(+)

The calculation of the actual number of
HD → HD(+) from the data is straight forward since
it cannot be confused with the other reactions. However, the other reactions are
coupled. When we detect an D(+) it could be from an HD → H + D(+) reaction or a
HD → H(+) + D(+) reaction in which we lost the H(+). Furthermore, when we detect
an H(+) and a D(+), that could be a HD → H(+) + D(+) reaction or a random
coincidence between a HD → H + D(+) reaction and a HD → H(+) + D reaction. Thus,
we arrive the following equations, where M(x) the measured counts of a species x
and N(x) is the actual number of counts of a species x:

Let ε be the detector efficiency and τ be the
likelihood of a random coincidence.

M[HD(+)] = εN[HD(+)]

M[H(+)] = εN[H(+) + D] + N[H(+) + D(+)]ε(1-ε)

M[D(+)] = εN[H + D(+)] + N[H(+) + D(+)]ε(1-ε)

M[H(+) + D(+)] = ε2N[H(+) + D(+)] +
τN[H(+) + D]N[H + D(+)]

We then apply several iterations to the coupled
equations with the first assumption being that τ = 0, since in reality τ is very
small.

1st iteration:

M[H(+)] = εN0[H(+) + D] + N0[H(+)
+ D(+)]ε(1-ε)

M[D(+)] = εN0[H + D(+)] + N0[H(+)
+ D(+)]ε(1-ε)

M[H(+) + D(+)] = ε2N0[H(+)
+ D(+)]

Solve for N0[H(+) + D], N0[H
+ D(+)], and N0[H(+) + D(+)].

2nd iteration:

M[H(+)] = εN1[H(+) + D] + N1[H(+)
+ D(+)]ε(1-ε)

M[D(+)] = εN1[H + D(+)] + N1[H(+)
+ D(+)]ε(1-ε)

M[H(+) + D(+)] = ε2N1[H(+)
+ D(+)] + τN0[H(+) + D]N0(H + D(+)]

Solve for N1[H(+) + D], N1[H
+ D(+)], and N1[H(+) + D(+)].

3rd iteration:

M[H(+)] = εN2[H(+) + D] + N2[H(+)
+ D(+)]ε(1-ε)

M[D(+)] = εN2[H + D(+)] + N2[H(+)
+ D(+)]ε(1-ε)

M[H(+) + D(+)] = ε2N2[H(+)
+ D(+)] + τN1[H(+) + D]N1(H + D(+)]

Solve for N2[H(+) + D], N2[H
+ D(+)], and N2[H(+) + D(+)].

Repeat the process until the solutions stop
changing. Again, with isotopic water, the process is much more complicated, but
the same general approach applies.

Finally, contamination is present in the target and this contamination
appears in the data. The contamination mainly consists of the other isotopic
water molecules from previous runs, and N2(+) and O2(+) from the atmosphere. To
remove the contamination from air, a run was conducted with only air. We
decoupled the air data and then removed it from the samples by scaling the ratio
of counts in a specific channel (i.e. m/q value, or values for pairs and
triples) to the counts in the O2(+) channel. We performed a similar scaling
procedure with removing the isotopic water contamination, but in a more precise
iterative method.

Problems in Correction Process

Unfortunately, all did not go as planned in the correction process. We first
examined the 4 MeV protons on isotopic water runs. H2O and D2O decoupled
perfectly within Itzik's program, but HDO did not. From that point, we attempted
two correction methods. First, we simply removed contamination from the data
without decoupling and compared our results to previous analysis of the data by
Max. This showed that while some purely coincidental channels remained within
the data, on channels that were realistic, the data was very close to Max's
previous analysis. Furthermore, the results of combining the coupled HDO file
with the decoupled H2O and D2O data allowed for the iterative method to
continue. This method produced results that are generally closer to Max's
previous analysis. Since the coupled method appears to have worked just as well
as decoupling, we applied the coupled method to the 1 MeV data. We have no check
for the results of this method since the 1 MeV data defied decoupling in Max's
previous analysis. We present some of our brief analysis of this 1 MeV data in
the final presentation. Unfortunately, due to time constraints we were unable to
include error.